Worksheet: Displacement Vectors

In this worksheet, we will practice calculating displacements in vector component form and calculating the resultant displacement of multiple linear displacements.

Q1:

A cyclist rides 5.0 km east,
then 10.0 km at an angle
20∘ west of north.
From this point she rides 8.0 km west.
What is the final displacement from where the
cyclist started? Consider east to correspond
to positive horizontal displacement and north to correspond to positive vertical displacement.

A(−1.6+4.3)ij km

B(−5.1+12)ij km

C(−6.4+9.4)ij km

D(−4.8+2.6)ij km

E(−8.2+8.4)ij km

Q2:

A pedestrian walks 6.00 km
east and then 13.0 km north.

Find the pedestrian’s resultant displacement.

At what angle north of east does the pedestrian walk?

Q3:

Suppose you walk 18.0 m
straight west and then 25.0 m straight north.

What is the distance from your starting point to where you are now?

What is the compass direction of a line connecting your starting point to your final position?

A38.8∘ west of north

B35.8∘ west of north

C42.9∘ west of north

D26.9∘ west of north

E32.1∘ west of north

Q4:

A particle located initially at (1.5+4.0)jk m
undergoes a displacement of (2.5+3.2−1.2)ijk m.
What is the final position of the particle?

A(4.0+4.0+3.2)ijk m

B(−3.3+8.3+2.0)ijk m

C(5.2+1.7−2.9)ijk m

D(3.7+2.2+4.8)ijk m

E(2.5+4.7+2.8)ijk m

Q5:

The 18th hole at Pebble Beach Golf Course is a dogleg to the left, with a length of 496.0 m. A golfer hits his tee shot a distance of 300.0 m, corresponding to a displacement Δ=300.0rim, and hits his second shot 189.0 m with a displacement Δ=(172.0+80.3)rijm from the second shot. What is the final displacement of the golf ball from the tee? Assume that the fairway off the tee corresponds to the positive 𝑥-direction.

A(472.0−80.3)ij m

B(472.0+80.3)ij m

C(128.0+80.3)ij m

D(300.0+200.0)ij m

E(−472.0+80.3)ij m

Q6:

A man walks from his cabin to the lake in a straight line for a distance of
5.0 kilometres, as shown in the following figure.

Determine the man’s displacement directly to the east.

Determine the man’s displacement directly to the north.

The man walks back from the lake to his cabin, never traveling in any direction other
than directly west or directly south. What is the minimum distance the man must
walk?

Q7:

A woman walks from her cabin to the lake in a straight line for a distance of
5.0 kilometers, as shown in the following figure.

Determine the east component of her displacement vector.

Determine the north component of her displacement vector.

How many more kilometers would she have to
walk if she walked along the component displacements?

Q8:

The F-35B is a short-takeoff and vertical landing fighter jet. An F-35B takes off vertically upward to a height of 20.00 m,
while still facing horizontally. The fighter then follows a flight path that is angled at
30.00∘ above a line parallel to the ground,
20.00 m vertically above the ground.
The fighter flies for a distance of 20.00 km along this trajectory. What is the fighter’s final displacement?

A1.570×10+1.025×10ij m

B1.572×10+1.020×10ij m

C1.057×10+1.205×10ij m

D1.753×10+1.005×10ij m

E1.732×10+1.002×10ij m

Q9:

Aaron Rodgers throws a football at 20.0 m/s
to his wide receiver, who runs straight down the field at 9.40 m/s
for 20.0 m. If Aaron throws the football when the wide receiver has
reached 10.0 m, what angle does Aaron have to launch the ball so the
receiver catches it at the 20.0 m mark?

Q10:

An airplane is initially at a point 𝑃. The airplane flies 32.0 km in a straight line, traveling 35.0∘ south of west, where it arrives at point 𝑃. A helicopter also flies from 𝑃 to 𝑃, but the helicopter initially travels in a direction 45.0∘ south of west, turning once to rotate its direction of travel by 90.0∘.

Find the distance that the airplane would need to fly directly west from 𝑃 to have the same westward displacement as the westward displacement from 𝑃 to 𝑃.

Find the distance that the airplane would need to fly directly south from 𝑃 to have the same southward displacement as the southward displacement from 𝑃 to 𝑃.

Find the distance that the helicopter travels south of west.

Find the distance that the helicopter travels west of north.

Q11:

In an attempt to escape a desert island, a castaway builds a raft and sets out to sea. The
wind shifts a great deal during the day, causing the castaway to lose control of her raft. A
search party used satellite images to track her movement from the point where she left the
island. Meanwhile, a rescue party traveled to intercept her. The following movements were
tracked and relayed to the rescue party:
45.0∘ north of west for
2.50 km, then
60.0∘ south of east for
4.70 km, then 2.50∘ south of west
for 5.10 km, then 5.00∘ east of north
for 1.70 km, then 55.0∘ south of west
for 7.20 km, and finally
10.0∘ north of east for
2.80 km. At this point, the castaway
was rescued.

What was the magnitude of the displacement of the castaway from the point she left the
island to the point where she was rescued?

What was the direction of the castaway’s displacement from the point she left the
island to the point where she was rescued?

A59.5∘ south of east

B11.3∘ north of east

C37.8∘ south of east

D43.5∘ south of east

E63.5∘ south of east

Q12:

A delivery man starts at the post office,
drives 40 km north, then
20 km west,
then 60 km northeast,
and finally 50 km north to stop for lunch.

Find the magnitude of the delivery man’s net displacement vector.
Give your answer to a precision of 3 significant figures.

Find the angle north of east made by the delivery man’s net displacement vector.
Give your answer to a precision of 2 significant figures.

Q13:

In an attempt to escape a desert island, a castaway builds a raft and sets out to sea. The wind shifts a great deal during the day and he is blown along the following directions: 2.50 km
and north of west, then 4.70 km
and south of east, then 1.30 km
and south of west, then 5.10 km
straight east, then 1.70 km and east of north, then 7.20 km
and south of west, and finally 2.80 km
and north of east.

Find the castaway’s final distance from the island.

Find the compass direction that this distance is in.

A48.5∘ south of east

B19.2∘ north of east

C63.5∘ south of east

D22.1∘ west of north

E13.5∘ west of south

Q14:

Starting at the island of Moi in an unknown archipelago, a fishing boat makes a round trip with two stops
at the islands of Noi and Poi. It sails from Moi for
4.76 nautical miles
(nmi) in a direction 37.0∘
north of east to Noi. From Noi, it sails 69.0∘
west of north to Poi. On its return leg from Poi, it sails
28.0∘ east of south.

Hint: 1=1,852nmim.

What distance does the boat sail between Noi and Poi?

What distance does the boat sail between Poi and Moi?

Q15:

An adventurous dog strays from home. The dog runs three blocks east, then two blocks north, then one block east, then one block north, and finally two blocks west. The length of each block is 100 yards. Take the positive 𝑥-axis as eastward.

Find the magnitude of the dog’s net displacement vector.

Find the direction of the dog’s net displacement vector.

A66.2∘ north of east

B39.2∘ north of east

C58.5∘ north of east

D56.3∘ north of east

E48.1∘ north of east

Q16:

You start a drive that is 7.50 km long from a point 𝑃,
moving in a straight line in a direction
15.0∘ east of north, and arrive at a point 𝑃.

Find the distance you would have to drive straight east from
𝑃 to be able to reach 𝑃 driving only straight north.

Find the distance you would have to drive straight north from
𝑃 to be able to reach 𝑃 driving only straight east.

Q17:

The position of a particle changes from rij=(1.3+4.2) cm to
rij=(−3.2+1.4) cm.
What is the displacement from r to r?

A(−1.9+5.6)ij cm

B(4.5+2.8)ij cm

C(−9.0−3.6)ij cm

D(1.9−5.6)ij cm

E(−4.5−2.8)ij cm

Q18:

A blue bottle fly lands on a sheet of graph paper at a point located 10.0 cm to the
right of its left edge and 8.0 cm above its bottom edge and walks slowly to a point
located 5.0 cm from the left edge and 5.0 cm from the bottom edge. A rectangular
coordinate system has its origin at the lower left-side corner of the sheet of
paper. Express the displacement vector of the fly due to its walking motion in this
coordinate system.

A(5.0+3.0)ij cm

B(−3.0−5.0)ij cm

C(−2.0−5.0)ij cm

D(2.0−3.0)ij cm

E(−5.0−3.0)ij cm

Q19:

A diver explores a shallow reef off the coast of Belize. She initially swims 90.0 m north, makes a turn to the east and continues for 200.0 m, then follows a big grouper for 80.0 m in the direction 30.0∘ north of east. In the meantime, a local current displaces her by 150.0 m south. The current then stops.

In what direction should she now swim to come back to the point where she started?

A4.64∘ south of west

B2.20∘ south of west

C2.10∘ north of west

D1.19∘ north of west

E3.12∘ south of west

What distance should she now swim to come back to the point where she started?

Q20:

The displacement vector of a blue bottle fly walking on a sheet of graph paper is Dij=(−5.00−3.00)cm.

What is the magnitude of the fly’s displacement?

At what angle below the negative 𝑥-axis is the fly’s displacement directed?

Q21:

A skater glides along a circular path of radius 5.00 m in clockwise direction. He coasts around one-half of the circle, starting from its westernmost point.

Find the magnitude of his displacement vector.

Find the distance that he skated.

What is the magnitude of his displacement vector when he skates all the way around the circle and comes back to the westernmost point?

Q22:

A soldier runs a distance of 20 m to the west. In a coordinate system where north corresponds to positive
𝑥-axis displacement and east corresponds to positive 𝑦-axis
displacement, what is the soldier’s displacement vector?

A20j m

B2j m

C−20j m

D10j m

E−10j m

Q23:

In the control tower at a regional airport, an air traffic controller monitors two aircraft as their positions change with respect to the control tower. One plane is a cargo carrier Boeing 747 and the other plane is a Douglas DC-3.

The Boeing is at an altitude of 2,500 m, climbing at 10.00∘ above the horizontal, and moving 30.00∘
north of west.

The DC-3 is at an altitude of 3,000 m, climbing at 5.000∘ above the horizontal, and cruising directly west.

Find the displacement vector relative to the control tower of the Boeing 747.

A(20.22+5.323+2.500)ijk km

B(15.02+9.371+3.000)ijk km

C(12.88+4.316+3.000)ijk km

D(12.27+7.089+2.500)ijk km

E(10.45+6.661+2.500)ijk km

Find the displacement vector relative to the control tower of the Douglas DC-3.

A(−0.3263+0.000+2.500)ijk km

B(−0.1189+1.127+3.000)ijk km

C(−0.4721+0.000+3.000)ijk km

D(−0.2274+1.324+2.500)ijk km

E(−0.2620+0.000+3.000)ijk km

Find the distance between the planes at the moment the air traffic controller makes a note about their positions.

Q24:

A mouse pointer on the display monitor of a computer at its initial position is at point
(6.0 cm,
1.6 cm) with respect to
the screen’s lower left-side corner. The pointer is moved to an icon located at point
(2.0 cm,
4.5 cm), giving the
pointer a displacement vector r.

What is the magnitude of r?

At what angle below the positive 𝑥-axis is the direction of
r?

Q25:

Madison set out from her home to deliver flyers for her yard sale, traveling due
east along her street. In 540 seconds of walking, Madison moved
0.500 km and also used up
her supply of flyers. Madison retraced her steps back to her house to get more
flyers, taking another 540 s. After
picking up more flyers, Madison set off again on the same path as before,
continuing to post flyers from where she had left off, and kept distributing flyers
until she reached a point
1.000 km from her house. This
third leg of her trip took 810 seconds. After this point, Madison
turned back toward her house, heading west. After moving
1.750 km west in
1,500 seconds,
Madison stopped to rest. Assume that displacement east corresponds to positive
values.

What was Madison’s displacement from her house to the point where she
stopped to rest?

What was the magnitude of Madison’s displacement from her house to the
point where she stopped to rest?

What was Madison’s average velocity from the instant that she first left
her house to the instant that she stopped to rest?

What distance did Madison travel between first leaving her house and stopping
to rest?